1904 - 5 .] Tliree-line Determinants of a Six-by- Three Array. 367 
Of course in the sixty equations here implied every distinct 
equation is repeated four times ; for example, the equation 
00' - 11' - 44' - 77' = 0 occurs under each of the headings 00', 
IT, 44', 77'. The number of distinct equations is thus 15. 
(5) These fifteen equations are not all independent, the fact 
being that any one of the ten sets of six gives rise to all the 
remaining nine equations. Thus, taking the first set of six, viz. 
00' -11' -44' - 77 ' = 0 , 
00' -88' -22' -55 ' = 0 , 
00' -66' -99' -33 ' = 0 , 
00'-ll'-88'-66' = 6>, 
00' -44' -22' -99 ' = 0 , 
00' -77' -55' -33 ' = 0 , 
we can eliminate from pairs of them the nine binomials 
00'- 11', 00' -44', 00' -77', 
00' -22', 00' -55', 00' -88', 
00' -33', 00' -66', 00' -99', 
thus obtaining nine other equations of the same form, which are 
the nine in question. It is thus seen that the connecting 
equations will be better viewed as statements of the equality of 
binomials ; and the theorem which this view leads to is that 
either the sum or the difference of any two of the products 
00', IT, .... is expressible in two ways as the sum or difference 
of other two. The forty-five possible binomials may be arranged 
as follows to show these equalities : — 
00' -IT = 77' + 44' = 88' + 66' | 
00' - 22' = 88' + 55' = 99' + 44' > 
00' - 33' = 99' + 66' = 77' + 55' ) 
00' - 44'= IT + 77' = 22' + 99' j 
00' - 55' = 22' + 88' = 33' + 77' 
00' - 66' = 33' + 99' = 1 T + 88' ) 
00' - 77' = 1 T + 44' = 33' + 55 ' ) 
00' - 88' = 22' + 55' = 1 T + 66' > 
00' -99' = 33' + 66' = 22' + 44') 
